Algebraic constructive quantum field theory: Integrable models and deformation techniques

Several related operator-algebraic constructions for quantum field theory models on Minkowski spacetime are reviewed. The common theme of these constructions is that of a Borchers triple, capturing the structure of observables localized in a Rindler wedge. After reviewing the abstract setting, we discuss in this framework i) the construction of free field theories from standard pairs, ii) the inverse scattering construction of integrable QFT models on two-dimensional Minkowski space, and iii) the warped convolution deformation of QFT models in arbitrary dimension, inspired from non-commutative Minkowski space. 1 Algebraic constructions of quantum field theories 1 2 Operator-algebraic constructions based on wedge algebras 5 2.1 Wedges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Wedge algebras and Borchers triples . . . . . . . . . . . . . . . . . 9 2.3 Standard pairs and free field theories . . . . . . . . . . . . . . . . . 13 2.4 Relative commutants of wedge algebras . . . . . . . . . . . . . . . 17 3 Integrable models and inverse scattering theory 21 3.1 Factorizing S-Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2 Construction of Borchers triples from two-particle S-matrices . . 26 3.3 Regular S-matrices and the modular nuclearity condition . . . . 33 3.4 Discussion of the constructed models . . . . . . . . . . . . . . . . . 35 3.5 Massless models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4 Deformations of quantum field theories by warped convolution 41 4.1 Warped convolutions and Rieffel deformations . . . . . . . . . . . 42 4.2 Borchers triples and warped convolutions . . . . . . . . . . . . . . 45 References 49 ∗Institut für Theoretische Physik, Universität Leipzig, [email protected].